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Multivariate Description

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Multivariate Description

> summary(gulls.pca2)

Importance of components:

Comp.1 Comp.2 Comp.3 Standard deviation 1.8133342 0.52544623 0.47501980 Proportion of Variance 0.8243224 0.06921464 0.05656722 Cumulative Proportion 0.8243224 0.89353703 0.95010425

> gulls.pca2$loadings

Loadings:

Comp.1 Comp.2 Comp.3 Comp.4Weight -0.505 -0.343 0.285 0.739Wing -0.490 0.852 -0.143 0.116Bill -0.500 -0.381 -0.742 -0.232H.and.B -0.505 -0.107 0.589 -0.622

There are (at least) two models:-

- Linear - species increase or decrease along the environmental gradient
- Unimodal - species rise to a peak somewhere along the environmental gradient and then fall again

- Environmental gradients are inferred from species data alone
- Three methods:
- Principal Component Analysis - linear model
- Correspondence Analysis - unimodal model
- Detrended CA - modified unimodal model

standard

biodynamic& hobby

nature

- Use single responses in linear models of environmental variables
- Use axes of a multivariate dimension reduction technique as responses in linear models of environmental variables
- Constrain the multivariate dimension reduction into the factor space defined by the environmental variables

- Start with all the variability in the response variables.
- Replace the original observations with their fitted values from a model employing the environmental variables as explanatory variables (discarding the residual variability).
- Carry our gradient analysis on the fitted values.

- Unconstrained ordination axes correspond to the directions of the greatest variability within the data set.
- Constrained ordination axes correspond to the directions of the greatest variability of the data set that can be explained by the environmental variables.

- Environmental gradients are constructed from the relationship between species environmental variables
- Three methods:
- Redundancy Analysis - linear model
- Canonical (or Constrained) Correspondence Analysis - unimodal model
- Detrended CCA - modified unimodal model

Similarity approaches

example

Continuous data:height

Categorical data

ordered (nominal):growth rate

very slow, slow, medium, fast, very fast

not ordered:fruit colour

yellow, green, purple, red, orange

Binary data:fruit / no fruit

We define a similarity between units – like the correlation between continuous variables.

(also can be a dissimilarity or distance matrix)

A similarity can be constructed as an average of the similarities between the units on each variable.

(can use weighted average)

This provides a way of combining different types of variables.

A

B

A

B

relevant for continuous variables:

Euclidean

city block or Manhattan

(also many other variations)

0,0

1,0

0,1

1,1

0,0

1,0

0,1

1,1

simple matching

count if both units 0 or both units 1

Jaccard

count only if both units 1

(also many other variants, eg Bray-Curtis)

simple matching can be extended to categorical data

Distance/Dissimilarity can be used to:-

- Explore dimensionality in data using Principal coordinate analysis (PCO or PCoA)
- As a basis for clustering/classification

NMDS maps the observed dissimilarities onto an ordination space by trying to preserve their rank order in a low number of dimensions (often 2) – but the solution is linked to the number of dimensions chosen

it is like a non-linear version of PCO

define a stress function and look for the mapping with minimum stress

(e.g. sum of squared residuals in a monotonic regression of NMDS space distances between original and mapped dissimilarities)

need to use an iterative process, so try with many different starting points and convergence is not guaranteed

used to compare graphically two separate ordinations

Grouping methods

Cluster Analysis

- hierarchical
- divisive
- put everything together and split
- monothetic / polythetic

- agglomerative
- keep everything separate and join the most similar points (classical cluster analysis)

- divisive
- non-hierarchical
- k-means clustering

Single linkage or nearest neighbour

finds the minimum spanning tree:

shortest tree that connects all points

- chaining can be a problem

Complete linkage or furthest neighbour

- compact clusters of approximately equal size.
- (makes compact groups even when none exist)

Average linkage methods

- between single and complete linkage

Basically you just approach this in the same way as for multiple regression – so there are the same issues of variable selection, interactions between variables, etc.

However the basis of any statistical tests using distributional assumptions are more problematic, so there is much greater use of randomisation tests and permutation procedures to evaluate the statistical significance of results.

Some examples

Part of Fig 4.

Positive Matrix Factorisation

FROM: http://cires.colorado.edu/jimenez-group/UsrMtgs/UsersMtg8/Ulbrich_PMF_Cautions.pdf